PROVE that 7+2√5 is irrational
Answers
Let assume that 7-2root3 be rational and rational numbers are in the form of p/q form.
\begin{lgathered}7 - 2 \sqrt{3} = \frac{p}{q} \\ 7 - \frac{p}{q} = 2 \sqrt{3} \\ \frac{7q - p}{q} = 2 \sqrt{3} \\ \frac{7q - p}{2q} = \sqrt{3}\end{lgathered}
7−2
3
=
q
p
7−
q
p
=2
3
q
7q−p
=2
3
2q
7q−p
=
3
We know that root 3 is irrational and not p/q form. It contradicts the statement that it is irrational as it is p/q form.
Subtraction and multiplication of irrational number form irrational number so
7-2root3 is irrational.
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Let 7+2/5 is rational
It must be in form of a/b
a/b=7+2/5
a-7b/b=2/5
/5=a-7b/2b
/5is rational( because a-7b/2b is rational).
its contradict that /5 is irrational.
Hence our supposition was wrong
7+2/5 is irrational.